In this undergraduate thesis, we discuss the two-body quantum mechanical problem of two ultra- cold bosonic atoms at low energy limit, interacting through a square potential with and without an external confinement. The two-body s-wave scattering length is introduced in terms of the radial wavefunction. In free space with no confinement, the interaction-scattering length relation is calculated and compared in 3- and 1-dimensions; we show the recurring resonance regions where the scattering length diverges and the existence of dimer bound state with positively large scattering length. Lastly, in the case of a 3-dimensional isotropic harmonic confinement, again under the ultracold limit, we obtain the universal (s-state) energy spectrum of the two atoms in terms of the scattering length.
We solve the three-body problem of a quasi-one-dimensional ultracold Fermi gas with parabolic confinement length a (perpendicular) and 3D scattering length a. On the two-body level, there is a Feshbach-type resonance at a (perpendicular)/a approximately 1.46, and a dimer state for arbitrary a (perpendicular)/a. The three-body problem is shown to be universal, and described by the atom-dimer scattering length a(ad) and a range parameter b(ad). In the dimer limit a (perpendicular)/a>>1, we find a repulsive zero-range atom-dimer interaction. For a (perpendicular)/a<<-1, however, the potential has long range, with a(ad)>0 and b(ad)>>a(ad). There is no trimer state, and despite a(ad)=0 at a( perpendicular)/a approximately 2.6, there is no resonance enhancement of the interaction.
1. Highlight “short-ranged,” “dilute,” and “low-energy” as three main features of interactions between ultracold atoms. 2. Introduce the important concept of the phase shift. 3. Introduce the
s-wave scattering length as a universal parameter describing the low-energy interaction between ultracold atoms. 4. Discuss the relation between divergent scattering length, low-energy bound state, and jump of phase shift. 5. Discuss the relation between the scattering length and the scattering amplitude. 6. Discuss under what condition a positive scattering length describes repulsive interaction. 7. Discuss the condition when an algebraically decayed potential can be treated as a finite range one. 8. Introduce two types of zero-range single-channel potentials to capture the universal low-energy
s-wave interaction between ultracold atoms. 9. Introduce the concept of renormalization condition and renormalizable theory. 10. Discuss how the spin rotational symmetry imposes constraints on interaction form for both alkali-metal and alkaline-earth-metal atoms. 11. Introduce Feshbach resonance as an important tool to tune scattering length. 12. Compare the two-channel Feshbach resonance with the single-channel shape resonance and compare the wide and the narrow resonances. 13. Introduce a zero-range two-channel model. 14. Introduce the confinement-induced resonance to tune interaction strength by an external potential. 15. Summarize three key conditions for a Feshbach resonance, and unify the optical Feshbach resonance, the orbital Feshbach resonance, and the confinement-induced resonance all in terms of these three conditions. 16. Introduce the Efimov effect as an important three-body effect at the vicinity of the two-body scattering resonance. 17. Highlight the symmetry aspect of the Efimov effect. 18. Discuss various connections between few-body and many-body physics. 19. Illustrate that few-body calculation can be used to determine properties of many-body systems by using high-temperature expansion as an example.
We study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale ${a}_{\ensuremath{\perp}}$. For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length
a and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths
aad and
bad. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find
bad=0 and
aad is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal:
aad and
bad depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter
R∗ related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.
This thesis presents a simple theoretical model of a Feshbach resonance between two colliding ultracold atoms and experimental progress towards the observation of such a resonance in Rb85. On the theoretical side, the inter-atomic scattering potential is approximated by a spherical square well potential. The potential supports a number of bounds states whose energy depends on the width and depth of the well. In the low energy limit, only s-wave scattering is present, and the associated scattering length for the process diverges if there is a bound state at the dissociation limit: this is the hallmark of a Feshbach resonance. The model also shows that the Feshbach resonance can be shifted to higher scattering energies by adjusting the well depth, indicating the presence of a quasi-bound state above the dissociation limit. On the the experimental side, a crossed dipole trap has been developed and used to trap ultracold rubidium atoms. The performance of the trap is characterized, and the thesis describes work towards upgrading it for an experiment to observe a Feshbach resonance in the F=2, m_F=-2 hyperfine ground state of Rb85.
We investigate the scattering and two-body bound states of two ultracold atoms in a quasi-two-dimensional (quasi-2D) confinement, with the confinement potential being an infinitely deep square well (box potential) in the transverse (
z) direction, and the motion of the atoms in the $x\text{\ensuremath{-}}y$ plane being free. Specifically, we calculate the effective 2D scattering length and 2D effective range parameter of the low-energy scattering, as well as the energy and the transverse-excited-mode probability of the bound states. Comparing these results with those obtained under a harmonic transverse confinement potential, which has the same characteristic length as the box potential, we find that in most of the cases the 2D effective range parameter for the box confinement is approximately 0.28 of the one for the harmonic confinement. Moreover, the transverse-excited-mode probability of the bound states for the box confinement is also much lower than the one for the harmonic confinement. These results suggest that the transverse excitation in the box confinement is notably weaker than the one in the harmonic confinement with the same characteristic length. Therefore, achieving quasi-2D ultracold gases well described by pure-2D effective models, particularly those with 2D contact interaction, is more feasible through box confinement. Our results are helpful for the quantum simulation of 2D many-body physics with ultracold atoms, e.g., the suppression of 2D effective range parameter may lead to an enhancement of quantum anomaly in two-dimensional Fermi gases. Additionally, our calculation method is applicable to the two-body problems of ultracold atoms in other types of quasi-2D confinements.
The two-body scattering amplitude and energy spectrum of confined ultracold atoms are of fundamental importance for both theoretical and experimental studies of ultracold-atom physics. For many systems, one can efficiently calculate these quantities via the zero-range Huang-Yang pseudopotential (HYP), in which the interatomic interaction is characterized by the scattering length
a. Furthermore, when the scattering length is dependent on the kinetic energy ${\ensuremath{\varepsilon}}_{\mathrm{r}}$ of two-atom relative motion, i.e., $a=a({\ensuremath{\varepsilon}}_{\mathrm{r}})$, the results are applicable for a broad energy region. However, when the free Hamiltonian of atomic internal state (e.g., the Zeeman Hamiltonian) does not commute with the interatomic interaction, or the center-of-mass (c.m.) motion is coupled to the relative motion, the generalization of this technique is still lacking. In this work we solve this problem and construct a reasonable energy-dependent multichannel HYP, which is characterized by a ``scattering length operator'' ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$, for the above complicated cases. Here, ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ is an operator for atomic internal states and c.m. motion, and depends on both the total two-atom energy and the external field as well as the trapping parameter. The effects from the internal-state or c.m.-relative motion coupling can be self-consistently taken into account by ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$. We further show a method based on the quantum defect theory, with which ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ can be analytically derived for systems with van der Waals interatomic interaction. To demonstrate our method, we calculate the spectrum of two ultracold fermionic alkaline-earth-like atoms [in electronic
1S0 $(|g\ensuremath{\rangle})$ and
3P0 $(|e\ensuremath{\rangle})$ states, respectively] confined in an optical lattice. By comparing our results with the recent experimental measurements for two
173Yb atoms and two
171Yb atoms, we calibrate the scattering lengths ${a}_{\ifmmode\pm\else\textpm\fi{}}$ with respect to antisymmetric and symmetric nuclear-spin states to be
a+=2012(19)a0 and ${a}_{\ensuremath{-}}=193(4){a}_{0}$ for
173Yb, and
a+=232(3)a0 and ${a}_{\ensuremath{-}}=372(1){a}_{0}$ for
171Yb.